{"id":5192,"date":"2014-03-27T06:00:21","date_gmt":"2014-03-26T21:00:21","guid":{"rendered":"http:\/\/fujiitoshiki.com\/improvesociety\/?p=5192"},"modified":"2014-09-10T16:37:53","modified_gmt":"2014-09-10T07:37:53","slug":"post-5192","status":"publish","type":"post","link":"https:\/\/www.fujiitoshiki.com\/improvesociety\/?p=5192","title":{"rendered":"∇\u3092\u6709\u3059\u308b\u5f0f"},"content":{"rendered":"

\u3000\u4eee\u306b \\bold{A}, \\bold{B}, U \u304a\u3088\u3073 V \u306e\u504f\u5fae\u5206\u304c\u5b58\u5728\u3059\u308b\u3068\u4eee\u5b9a\u3059\u308b\u3068<\/p>\n

    \n
  1. \\displaystyle \\nabla(U + V) = \\nabla U + \\nabla V \u307e\u305f\u306f grad(U + V) = grad U + grad V<\/li>\n
  2. \\displaystyle \\nabla\\cdot(\\bold{A} + \\bold{B}) = \\nabla\\cdot\\bold{A} + \\nabla\\cdot\\bold{B} \u307e\u305f\u306f div(\\bold{A} + \\bold{B}) = div\\bold{A} + div\\bold{B}<\/li>\n
  3. \\displaystyle \\nabla\\times (\\bold{A} + \\bold{B}) = \\nabla\\times\\bold{A} + \\nabla\\times\\bold{B} \u307e\u305f\u306f curl(\\bold{A} + \\bold{B}) = curl\\bold{A} + curl\\bold{B}<\/li>\n
  4. \\displaystyle \\nabla\\cdot(U\\bold{A}) = (\\nabla U)\\cdot\\bold{A} + U(\\nabla\\cdot\\bold{A})<\/li>\n
  5. \\displaystyle \\nabla\\times(U\\bold{A}) = (\\nabla U)\\times\\bold{A} + U(\\nabla\\times\\bold{A})<\/li>\n
  6. \\displaystyle \\nabla\\cdot(\\bold{A}\\times\\bold{B}) = \\bold{B}\\cdot(\\nabla\\times\\bold{A}) - \\bold{A}\\cdot(\\nabla\\times\\bold{B})<\/li>\n
  7. \\displaystyle \\nabla\\times(\\bold{A}\\times\\bold{B}) = (\\bold{B}\\cdot\\nabla)\\bold{A} - \\bold{B}(\\nabla\\cdot\\bold{A}) - (\\bold{A}\\cdot\\nabla)\\bold{B} + \\bold{A}(\\nabla\\cdot\\bold{B})<\/li>\n
  8. \\displaystyle \\nabla(\\bold{A}\\cdot\\bold{B}) = (\\bold{B}\\cdot\\nabla)\\bold{A} + (\\bold{A}\\cdot\\nabla)\\bold{B} + \\bold{B}\\times(\\nabla\\times\\bold{A}) + \\bold{A}\\times(\\nabla\\times\\bold{B})<\/li>\n
  9. \\displaystyle \\nabla\\cdot(\\nabla U) \\equiv \\nabla^2U \\equiv \\frac{\\partial^2U}{\\partial x^2} + \\frac{\\partial^2U}{\\partial y^2} + \\frac{\\partial^2U}{\\partial z^2} \u306f U \u306e \u30e9\u30d7\u30e9\u30b9<\/em> \u3068\u547c\u3073\uff0c
    \n\\displaystyle \\nabla^2 \\equiv \\frac{\\partial^2}{\\partial x^2} + \\frac{\\partial^2}{\\partial y^2} + \\frac{\\partial^2}{\\partial z^2} \u306f \u30e9\u30d7\u30e9\u30b9\u6f14\u7b97\u5b50<\/em> \u3068\u547c\u3076\uff0e<\/li>\n
  10. \\displaystyle \\nabla\\times(\\nabla U) = 0. U \u306e\u52fe\u914d\u306e\u56de\u8ee2\u306f\u30bc\u30ed\u3067\u3042\u308b\uff0e<\/li>\n
  11. \\displaystyle \\nabla\\cdot(\\nabla\\times\\bold{A}) = 0. \\bold{A} \u306e\u56de\u8ee2\u306e\u767a\u6563\u306f\u30bc\u30ed\u3067\u3042\u308b\uff0e<\/li>\n
  12. \\displaystyle \\nabla \\times (\\nabla \\times \\bold{A}) = \\nabla (\\nabla \\cdot \\bold{A}) - \\nabla^2\\bold{A}<\/li>\n<\/ol>\n